Posts Tagged ‘Gödel’

About Morphology or How Alan Turing Made the Dream of Goethe Come True

Tuesday, November 17th, 2009

The Ancient Greeks believed that the images of waking life and dreams came from the same source, Morpheus (Μορφέας, Μορφεύς), “He who Shapes“.

The Science of the Shapes, Morphology, was created and named by Goethe in his botanical writings (“Zur Morphologie“, 1817).

Goethe used comparative anatomical methods, to discover a primal plant form that would contain all the others-the Urpflanze. Goethe being a Romantic Idealist hoped that Morphology would Unify Science and Art.

The Uhrplant shows itself also in the Lungs and Riversystems

The Uhrplant shows itself also in the Lungs and Riversystems

“The Primal Plant is going to be the strangest creature in the world, which Nature herself shall envy me. With this model and the key to it, it will be possible to go on forever inventing plants and know that their existence is logical”. Nature always plays, and from which she produces her great variety. Had I the time in this brief span of life I am confident I could extend it to all the realms of Nature – the whole realm“.

Goethe (wikipedia)

Goethe (wikipedia)

Hundred years later in the 1920s Goethe’s dream came true. Morphology moved outside Biology to other parts of Science due to the works of D’Arcy Thompson’s On Growth and Form, Oswald Spengler Morphology of History, Carol O. Sauer Morphology of Landscape, Vladimir Propp, Morphology of the Folktale and Alfred North Whitehead Process and Reality.

Goethe observed nature and reflected on similar structures. He believed that there was something behind this similarity, an archetypal plant.

According to Goethe the archetypal plant was the leaf (“While walking in the Public Gardens of Palermo it came to me in a flash that in the organ of the plant which we are accustomed to call the leaf lies the true Proteus who can hide or reveal himself in all vegetal forms. From first to last the plant is nothing but leaf“).

At this moment scientists know the reason why the leaf is the most important structure of the plant. It is a solar collector full of photosynthetic cells.

The energy of the sun provides the energy to transform water from the roots gathered by the leafs and carbon dioxide out of the air also gathered by the leafs, into sugar and oxygen. Plants are structures with many leaves. These leafs shield other leafs from collecting sunlight and water.

To solve this problem a plant has to optimize its structure to collect enough Sunlight and Water. The process of Optimization is not a Central Coordinated action. Every leaf tries to find the best place in the Sun on its own. This place determinates the growth of the next level of branches and leafs.

Goethe observed a pattern and deduced a structure, the leaf, the Uhrplanze. What Goethe really observed was not a Static Uhrplant but the Dynamic Process of the Branching of all kinds of leaves in all kinds of plants (Morpho-Genesis).

The leafs of the plants are not the main target of the morphogenesis of the plant. The visible External and the invisible Internal Forms or Organs are one of the many solutions of an equation with many variables and constraints. The optimal solution is reached by experimenting (“Nature always plays”).

Many solutions fail but some survive (Evolution of the Fittest). When a solution survives it is used as a Foundation to find new rules for more specific problems (Specialization). When the environment, the context, changes old rules have to be replaced by new rules (a Paradigm Shift).

The Fractal Geometry of Nature

The Fractal Geometry of Nature

New mathematical paradigms in the field of the Machines and Languages (Alan Turing, The Chemical Basis of Morphogenesis) and the Self-Referencial Geometry of Nature (Benoît Mandelbrot, The Fractal Geometry of Nature) have stimulated further investigation in the Field of Morphology.

In 1931, in a monograph entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems Gödel proved that it is impossible to define a theory that is both Self-Consistent and Complete. The paper of Gödel destroyed the ambitions of the Mathematicians at that time to define one theory that explains everything.

In 1936 Alan Turing produced a paper entitled On Computable Numbers. In this paper Alan Turing defined a Universal Machine now called a Turing Machine. A Turing machine contains an infinite tape that can move backwards and forwards and a reading/writing device that changes the tape. The Turing Machine represents every Theory we can Imagine.

Turing proved that the kinds of questions the machine can not solve are about its own Performance. The machine is Unable to Reflect about Itself. It needs another independent machine, an Observer or Monitor to do this.

It can be proved that Turing proved the so called Incompleteness Theorem and the Undecidability Theorem of Gödel in a very simple way.


The Eniac

In 1943 Turing helped to Crack the Codes of the Germans in the Second World War. At that time the first computers were build (Eniac, Collossus).

It was very difficult to Program a Computer. This problem was solved when Noam Chomsky defined the Theory of Formal Grammars in 1955 (The Logical Structure of Linguistic Theory).

When you want to define a Language you need two things, an Alphabet of symbols and Rules. The symbols are the End-Nodes (Terminals) of the Network of Possibilities that is produced when the Rules (Non-Terminals) are Applied. The Alphabet and the (Production- or Rewriting) rules are called a Formal Grammar.

If the Alphabet contains an “a” and a “p” the rules S→AAP, A→”a” and P→”p” produce the result “aap”. Of course this system can be replaced by the simple rule S→”aap”. The output becomes an infinite string when one of the rules contains a Self-Reference. The rules A→a and S→AS produce an Infinity String of “a’-s (“aaaaaaaaaaaaaaaaaa….”).

The system becomes more complicated when we put terminals and rules (non-terminals) on the Left Side. The System S→aBSc, S→abc, Ba→aB and Bb→bb produces strings like, “abc”, “aabbcc” and “aaabbbccc”. In fact it produces all the strings a**n/b**n/c**n with n>0.

The inventor of the theory of Formal Grammar, Chomsky, defined a Hierarchy of Languages. The most complex languages in his hierarchy are called Context-Dependent and Unrestricted. They represent complex networks of nodes.

A language where the left-hand side of each production rule consists of only a single nonterminal symbol is called a Context Free language. Context Free Languages are used to define Computer Languages. Context Free Languages are defined by a hierarchical structure of nodes. Human Languages are dependent on the context of the words that are spoken.

It is therefore impossible to describe a Human Language, Organisms, Organisations and Life Itself with a Context Free Computer Language.

Context Free Systems with very simple rule-systems produce natural and mathematical structures. The System A → AB, B → A models the Growth of Algae and the Fibonacci Numbers.

A Recognizer or Parser determinates if the output of a formal grammar is produced by the grammar. Parsers are used to check and translate a Program written in a Formal (Context Free) Language to the level of the Operating System of the Computer.

grammarRegular and Context Free Grammars are easily recognized because the process of parsing is linear (causal, step by step). The stucture of the language is a hierarchy.

The recognizer (now called a Push-Down Machine) needs a small memory to keep the books.

Context Dependent (L-systems) and Unrestricted Grammars are difficult to recognize or are not recognizable in practice because the parser needs a huge sometimes Infinite Memory or Infinite Time to complete its task.

To find the Context the Recognizer has to jump backwards and forwards through the infinite string to detect the pattern.

If the network loops the recognizer will Never Stop (“The Halting Problem“).

Turing proved that the Halting Problem is Undecidable. We will Never Know for Sure if an Unrestricted Grammar contains Loops.

The Rules and the Output of Unrestricted Grammars Change and never stop Changing. Our Reality is certainly Context Dependent and perhaps Unrestricted.

Parsing or Recognizing looks like (is similar with) the process of Scientific Discovery. A theory, a Grammar of a Context-Free Systems (“aaaaaaaaaaa…”) is recognizable (testable) in Finite Time with a Finite Memory. Theories that are Context Dependent or Unrestricted cannot be proved although the Output of the Theory generates Our Observation of Nature. In this case we have to trust Practice and not Theory.

cellular automata

A 3D Cellular Automaton

In 2002 the Mathematician Stephen Wolfram wrote the book A New Kind of Science.

In this book he tells about his long term Experiments with his own Mathematical Program Mathematica. Wolfram defined a System to Generate and Experiment with Cellular Automata.

Wolfram believes that the Science of the Future will be based on Trial and Error using Theory Generators (Genetic Algorithms). The big problem with Genetic Algorithms is that they generate patterns we are unable to understand. We cannot  find Metaphors and Words to describe the Patterns in our Language System.

This problem was adressed by the famous Mathematician Leibniz who called this the Principle of Sufficient Reason.

Leibniz believed that our Universe was based on Simple Understandable Rules that are capable of generating Highly Complex Systems.

It is now very clear that the Self-Referencial Structures, the Fractals, of Mandelbrot are the solution of this problem.

The Scientific Quest at this moment is to find the most simple Fractal Structure that is capable of explaining the Complexity of our Universe. It looks like this fractal has a lot to do with the Number 3.

It is sometimes impossible to define a structured process to recognize (to prove) a Grammar. Therefore it is impossible to detect the rules of Mother Nature by a Structured process. The rules of Mother Nature are detected by Chance just like Goethe discovered the Uhrplanze. Science looks a lot like (is similar with) Mother Nature Herself.

When a Grammar is detected it is possible to use this grammar as a Foundation to find new solutions for more specific problems (Specialization, Add More Rules) or when the system is not able to respond to its environment it has to Change the Rules (a Paradigm Shift). All the time the result of the System has to be compared with Mother Nature herself (Recognizing, Testing, Verification).

Turing proved that if Nature is equivalent to a Turing machine we, as parts of this machine, can not generate a complete description of its functioning.

In other words, a Turing machine, A Scientific Theory, can be a very useful tool to help humans design another, improved Turing Machine, A new Theory, but it is not capable of doing so on its own – A Scientific Theory, A System, can not answer Questions about Itself.

The solution to this problem is to Cooperate. Two or more (Human) Machines, A Group, are able to Reflect on the Other. When the new solution is found the members of the Group have to Adopt to the new solution to move on to a New Level of Understanding and drop their own Egoistic Theory.

Each of the individuals has to alter its Own Self and Adapt it to that of the Group. It is proved that Bacteria use this Strategy and are therefore unbeatable by our tactics to destroy them.

Turing proved that Intelligence requires Learning, which in turn requires the Human Machine to have sufficient Flexibility, including Self Alteration capabilities. It is further implied that the (Human) Machine should have the Freedom to make Mistakes.

Perfect Human Machines will never Detect the Patterns of Nature because they get Stuck in their Own Theory of Life.

The Patterns of Turing

The Patterns of Turing

The Only ONE who is able to Reflect on the Morphogenesis of Mother Nature is the Creator of the Creator of Mother Nature, The Void.

Gregory Chaitin used the theory of Chomsky and proved that we will never be able to understand  The Void.

The Void is beyond our Limits of Reason. Therefore the first step in Creation will always be  a Mystery.

At the end of his life (he commited suicide) Alan Turing started to investigate Morphology.

As you can see the Patterns of Alan Turing are created by combining many Triangels. The Triangel is called the Trinity in Ancient Sciences.

According to the Tao Tse King, “The Tao produced One; One produced Two; Two produced Three; Three produced All things”, which means that the Trinity is the Basic Fractal Pattern of the Universe.

In modern Science this pattern is called the Bronze Mean.

It generates so called Quasi Crystals and the Famous Penrose Tilings.

The Bronze Mean is represented by the Ancient Structure of the Sri Yantra (“Devine Machine”).

Goethe was not the real discoverer of Morphology. The knowledge was already there 8000 years ago.


About the Observer and Second Order Cybernetics

A PDF About the Morphology of Music.

The origins of life and context-dependent languages

A Website About the Morphology of Botanic Systems

A Website About the Morphology of Architectural Systems

A Plant Simulator using Morphology

About Intelligent Design

The Mathematical Proof of Gödel of the Existence of God

About Bacteria 

About the Bronze Mean

About the Trinity

Too Many Insights about IT-Architectures and IT-Strategy

Monday, November 19th, 2007

I have been responsible for IT-Architectures and IT-Strategy between 1984 and 1997. From 1997 until now I have reviewed many Architectures and Strategies when I was part of Meta Group (now Gartner).

An IT-Architecture is a System that describes the components of a Software System on an Abstract Level.

An IT-Strategy is a Process that contains Stages. In every Stage a new version of the IT-Architecture is implemented.

A Well-Formed IT-Strategy is able to Adapt the Old Version of the Architecture. When you’re Strategy has failed You have to Start All over Again.

There are two types of Systems. The first type contains Systems. The second type I call a Foundation. It is the level where we think “the Real Things are Happening”. The major problem is to define the Level of the Foundation.

If you look at your Own computer the Foundation lies deeper than you think. It depends on “What You Understand About a Computer“.

We could define the Foundation as the Operating System of Your Computer (most likely a Microsoft Operating System) but below this Foundation other Foundations are in Existence.

At a more abstract level you can see the real Problem. The problem is Containing.

If you use the Containing Metaphor you Never Stop but You Have to Stop Somewhere.

The level where you Stop is the level where you give the responsibility to An-Other. This can be an Organization, A Person or when we dig deep enough even Nature.

When you give the responsibility for something to an-other you have to Trust the Other and the Other has to take Care of You.

The reason why Architectures fail is that they are based on a Foundation that is not stable on the long term.

Suddenly somebody starts to tinker with the Foundation and suddenly everything goes wrong. This happens all the time. The others leave you alone and you have to take care of yourself.

A solution was to create a Foundation that was able to withstand every Change at the Lower level.

This layer was called Middleware. It is situated somewhere between the UP and the DOWN of all the Layers. History has proven that this solution is not helpful.

Everything changes all the time.

I want to give you a Model to understand the complexity of the problem. I use a Horizontal and a Vertical layer, A Matrix. Layered architectures can be mapped on a Matrix, a Cube or a higher dimensional Structure, N-Dimensional Space.

Every component can be described by a point that is connected to N-variables. The links between the components are lines connecting the points.

The first thing we could do is use one dimension to describe the “type” of the component (software, hardware, database, form, etc). If we create a picture of this dimension we have created a System Diagram. There are many types of System Diagrams invented. They were called a Method or a Modeling Language. Every new Method created a “Method War” because the Users always thought that their Method was the Best.

I participated in many activities of the IFIP (International Federation for Information Processing). We tried to find a way to find the “Best Method” to Improve the Practice. It was proven that there was no best method.

Many roads lead to Rome. See “Information Systems Design Methodologies: Improving The Practice“, T.W. Olle, H.G. Sol and A.A. Verrijn-Stuart, (Eds.), North-Holland.

At the end the major Method Wars ended with a Compromise, UML. Compromises are always the worst solution for a problem. UML is a very complicated method.

If we start the Diagram with an Event and we have chosen the right Modeling Language, now called a Programming Language, we are able to “simulate” the System or to “generate” the software. There are many Tools, Modelers, Simulators, Languages and Generators developed.

They also created wars (called Competition) between all kinds of Vendors. In the end many Tools were taken out of the market by the Vendors and the Users got stuck. They were unable to convert the old tools to the new tools or they simply did not take the time to do this. This problem is called the Legacy Problem.

The Market invented something to solve this problem called Reverse Engineering. Reverse Engineering proved to be a failure because the semantics, the meaning, of the software was gone.

When you deconstruct a car and you show an engineer all the parts he knows the parts belonged to a car. When you do this with something nobody ever knew it existed the only engineer that is capable to reconstruct the original is the engineer who constructed it.

When the software is old the original programmer is gone and nobody is able to understand what he was doing. Sometimes the software contains documentation. The programmer has written a Story about the Meaning of the Software. Programmers never took (and take) the time to do this.

I want to get back to N-Dimensional Space. I hope you understand that we when we use enough dimensions we are able to Model Everything.

We are also able to MAP Everything to Everything. Mapping or Converting could solve many problems.

There were Systems on the market that helped you to MAP one Structure to another Structure. An example was Rochade. I implemented Rochade when I was responsible for IT-Architectures and I know Rochade solved many problems with Legacy Systems.

Rochade used something called a Scanner or Parser. A Parser is a “piece of software” that translates a “piece of software” into another “piece of software”. It stored the data of the software (the meta-data) in a “general” format that could be translated to other formats.

When you program in a Software Language the code is translated to another Language. This happens many times until the software reaches the Lowest Level, The Processor or the CPU.

The CPU uses a Cycle to process a very simple language that consists of binary numbers. These numbers are real numbers or operations. The simplest operations are operations on a Set.

The whole concept of the CPU was invented by John von Neumann and is therefore named the Von Neumann Architecture.

The architecture of von Neumann has a big disadvantage called the Von Neumann bottleneck. The CPU is continuously forced to wait.

The Von Neumann Computer is Wasting Time and Energy.

An alternative is the Parallel Architecture. Parallel computing has recently become the dominant paradigm in computer architectures. The main reason is the Rise of the Internet.

The Rise of the Internet started the Fall of Centralized IT-Architectures and Centralized IT-Strategy.

At this moment we need another approach to Manage or Control the sofware-production of a big company.

This approach can be found in the Open Source Movement.

If we use the Matrix Approach we can answer interesting questions.

First I introduce a Rule.

When we increase the amount of Dimensions we are able to make every point and connection between a point Unique. If we do the opposite we are able to make every point and connection The Same.

When people talk about the Reuse of a Component we are looking for a Dimension were some points and their connections are the Same.

I hope you see that it is possible to Reuse Everything and to Reuse Nothing. The Choice is Yours.

This also the Practice in Software. When I discovered that the Year-2000 problem could lead to a disaster I started a research-project with the CWI. The CWI develop a very intelligent parser that could create Software-Maps.

When we studied the maps we saw that some pieces of software came back all the time. These were “citations”. One programmer invented a new construct and others reused the construct all the time. The major difference with the Theory of Reuse was that MANY Parts were the Same.

When you dig deep enough you always find “The Same”.

The CPU is handling binary codes and when you would come from another planet you would not understand why all these zero’s and 1′s are creating a DIVERSITY. They create a DIVERSITY because Something is Interpreting the Sequence. This Something is also a Program. This program uses a Theory about Languages. Most of the time it supposes a Context Free Language. A Context Free Language is a language where the interpretor always moves in one direction. It processes a List.

The Diversity a Computer Produces is based on one long List of Binary patterns. If we could analyze all the possible patterns we could find all the possible software-programs that could be build until Eternity. Because the binary codes can be mapped to the Natural Numbers. We need only one dimension (A line) to classify all the possible software-components in the World.

In 1931 Gödel’s stated the so called incompleteness theorems. He uses the Natural Numbers to prove that a part of our Human Reality cannot be described by a Computer Program.

There is something “left for us” that the Machines cannot take over. This part is related to the Emotions and the Imagination. We cannot Automate them. If we do this we stop Innovation and Commitment.

Now I want to come back to IT-Architectures.

When You start an IT-Architecture Out of Nothing you start with a Small Amount of Dimensions. The world looks very simple to you. When you add detail you have to increase the amount of dimensions. The effect of this is that Everything Changes. New Possibilities arise. If you go on using the Top-Down Approach you will move into a State of Huge Complexity. Always start in the Middle!

At a certain moment You have to move to Reality.This means Programming Software. At that moment You encounter something you never thought of. The Software Legacy!!!!!!!!!!!!

When you increase the Scope of your System and you leave the Boundaries of Your Company (the World of the Internet) the Complexity increases also.At that moment You encounter something you never thought of, Open Source. Millions of Possibilities arise and You don’t Know what to do!

Behind the Software are People. Some of them are creating Small Companies and they are doing things you’re company is also doing but they do it much cheaper and faster.

What to do?

If You Can’t Beat them Join Them.